Similarity Criteria

Two triangles are similar when they have the same shape but not necessarily the same size. Formally, triangle ABC is similar to triangle DEF — written △ABC ∼ △DEF — when all three pairs of corresponding angles are equal AND all three ratios of corresponding sides are equal. The common ratio of corresponding sides is called the scale factor, k. Similarity is a weaker condition than congruence: congruent triangles are always similar (with k = 1), but similar triangles need not be congruent. In practice you rarely have access to all six pieces of information (three angles and three sides) at once. The similarity criteria — AAA, SSS~, and SAS~ — each identify the minimum information needed to guarantee similarity. They mirror the congruence criteria (ASA, SSS, SAS) but replace 'equal lengths' with 'proportional lengths', and one of the three criteria relies entirely on angles rather than any side measurement. These criteria are powerful tools in deductive proof. Once you establish that two triangles are similar, you can immediately set up proportions among their sides to find unknown lengths, or deduce angle relationships across a complex figure. They underpin results throughout geometry — from the Pythagorean theorem's standard proof to trigonometric ratios and the properties of circles.

Key takeaways
  • AAA — Angle-Angle-Angle
  • SSS~ — Side-Side-Side Similarity
  • SAS~ — Side-Angle-Side Similarity
  • Applying the Criteria: Setting Up Proportions

AAA — Angle-Angle-Angle

If all three angles of one triangle are equal to the corresponding three angles of another triangle, the triangles are similar. Because the angle sum in any triangle is 180°, knowing just two pairs of equal angles automatically forces the third pair to be equal as well. This means the practical criterion is AA (Angle-Angle): if ∠A = ∠D and ∠B = ∠E, then ∠C = ∠F and △ABC ∼ △DEF. No side information is needed at all. The scale factor k can be anything. Proof sketch: place △DEF so that D coincides with A and DE lies along AB. Scale △DEF by k = AB/DE. The scaled triangle shares vertex A with △ABC and has its second vertex at B. Because corresponding angles are equal, the sides are parallel, so by the properties of parallel lines and transversals the triangles are equiangular and hence similar.

SSS~ — Side-Side-Side Similarity

If the three pairs of corresponding sides are all in the same ratio, the triangles are similar. Specifically, if AB/DE = BC/EF = CA/FD = k, then △ABC ∼ △DEF. The tilde (~) distinguishes this from the SSS congruence criterion. Unlike AAA, this criterion requires measuring all three sides of both triangles and checking that the three ratios are equal. If you find that two ratios are equal but the third is different, the triangles are NOT similar. Proof sketch: construct △D'E'F' congruent to △ABC. Then D'E' = AB = k·DE, E'F' = BC = k·EF, and F'D' = CA = k·FD. Dividing, D'E'/DE = E'F'/EF = F'D'/FD, so △D'E'F' ∼ △DEF by proportionality, meaning all their angles match. Since △D'E'F' ≅ △ABC, the angles of △ABC equal those of △DEF, confirming similarity.

SAS~ — Side-Angle-Side Similarity

If two pairs of corresponding sides are in the same ratio AND the included angles (the angles between those two sides) are equal, the triangles are similar. Formally: if AB/DE = AC/DF and ∠A = ∠D, then △ABC ∼ △DEF. The angle must be the INCLUDED angle — sandwiched between the two proportional sides. This is a crucial distinction: two proportional sides with a non-included equal angle do NOT guarantee similarity (analogous to the ambiguous case in congruence). Proof sketch: scale △DEF by factor k = AB/DE to get △D'E'F' where D'E' = AB and D'F' = AC. Since ∠D' = ∠D = ∠A, triangles ABC and D'E'F' have two equal sides with equal included angles, so they are congruent by SAS. Therefore △ABC ∼ △DEF with scale factor k.

Applying the Criteria: Setting Up Proportions

Once similarity is established, corresponding sides are proportional. Always write the correspondence carefully from the similarity statement. If △ABC ∼ △DEF with scale factor k, then: AB/DE = BC/EF = CA/FD = k. Equivalently (and often more useful), AB/BC = DE/EF (ratios within each triangle equal). A common error is mixing up which sides correspond. The order of letters in the similarity statement defines the correspondence: A↔D, B↔E, C↔F. Always identify corresponding vertices before writing any proportion.

The Key Equation

ABDE=BCEF=CAFD=k\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k
  • AB, BC, CA — Sides of triangle ABC (length units): The three side lengths of the first triangle, measured in any consistent unit.
  • DE, EF, FD — Sides of triangle DEF (length units): The three corresponding side lengths of the second triangle.
  • k — Scale factor (dimensionless): The constant ratio of any corresponding side of △ABC to the matching side of △DEF. If k > 1, △ABC is larger; if 0 < k < 1, △ABC is smaller.

Worked Example

Problem. In the diagram, △PQR has PQ = 6, QR = 8, PR = 10. Triangle STU has ST = 9, TU = 12, SU = 15. Determine whether the triangles are similar, state which criterion applies, and find the scale factor.

△PQR∼△STU,k=23\triangle PQR \sim \triangle STU,\quad k = \dfrac{2}{3}

Answer. △PQR ∼ △STU by SSS~ with scale factor k = 2/3.

Figure
Diagram
Similar Triangles: △ABC ∼ △DEF (scale factor k = 2)ABC65√61DEF1210√244∠A = ∠D = 90°, ∠B = ∠E, ∠C = ∠F | DE/AB = EF/BC = DF/AC = 2k = 2
Origins · 1 minEuclid of Alexandria and how this was discovered
EO
Euclid of Alexandria
-325–-265 · Greek (Hellenistic)

Euclid systematised all of Greek geometry in his 13-book work Elements (c. 300 BCE). Book VI of the Elements is devoted entirely to similar figures, where Euclid proved the foundational propositions on similar triangles — including the equivalents of AAA (Proposition VI.4), SSS~ (VI.5), and SAS~ (VI.6) — using the theory of proportion developed in Book V. His deductive framework set the standard for mathematical proof for over two thousand years. When King Ptolemy I of Egypt reportedly asked Euclid if there was a shortcut to learning geometry, Euclid is said to have replied: 'There is no royal road to geometry.' The same story is told about Alexander the Great and his teacher Menaechmus — suggesting it became a popular saying about the discipline required for mathematics.

  • Congruence Criteria (SSS, SAS, ASA, AAS) — Congruence criteria are the special case k = 1 of similarity criteria; understanding both together clarifies when shape alone vs. shape-and-size is determined.
  • Ratio and Proportion in Geometry — Setting up and solving proportions is the primary algebraic tool used once two triangles are proven similar.
  • The Pythagorean Theorem — The most elegant proof of the Pythagorean theorem uses AAA similarity: the altitude to the hypotenuse creates two triangles each similar to the original.
  • Trigonometric Ratios (sin, cos, tan) — Trig ratios are well-defined precisely because all right triangles with the same acute angle are similar (by AA), so the side ratios depend only on the angle.